This is quite a long post elaborating on a bit of work I’ve been doing with Greg Morley and his group, which was published in Scientific Reports. I hope it provides an accessible introduction to a fairly complicated set of experiments and simulations.

Background

Ablation is a treatment for some abnormal heart rhythms, which works by killing areas of the heart that we think are misbehaving and giving rise to abnormal rhythms. The idea behind ablation is that if you identify and kill a region of tissue that is allowing electrical signals to propagate ‘the wrong way’, then afterwards you will have created a non-conducting wall of scar tissue that prevents any abnormal electrical activity. As a treatment strategy, there’s clearly room for improvement; I hope we’ll soon look back on it as Dr McCoy does for dialysis in the scene in Star Trek IV. But at the moment, it’s the best we’ve got for certain conditions.

Usually, and ideally, you stop ablating when the arrhythmia terminates, so it is very effective in the short term. Unfortunately the success rate in the long term isn’t great – after a single round of atrial ablation people stay (for at least 3 years) arrhythmia free in only about 50% of cases1. So why do people re-lapse after ablation? Well it can be because electrical waves somehow get through the scar regions (see 1 for further references to this).

You also get similar scars forming after a heart attack – in this case the cardiac muscle itself doesn’t get a blood supply, and dies because of that, leaving a scar region. This damage leads to scars with larger border zones than in ablation, as in these border zones have low but not-zero blood supply, so you get semi-functioning tissue where some of the muscle cells (myocytes) survive, and others don’t. The problem now is the opposite to ablation – you’d like to restore conduction in these regions to get the rest of the heart beating as well as it can, but electrical activity is disturbed by the presence of these scar regions and their border zones, and arrhythmias become more likely (or even occur instantly in the case of big heart attacks).

Greg works on optical mapping for whole hearts in various species to study what happens around these scars. This is a clever technique where you put a dye into a tissue, it sticks to the membrane of cells, and when excited by an external light source, it emits light at an intensity dependent on the voltage across the cell membrane. So you can excite the dye with light at a certain wavelength, and record the light emitted at another wavelength in a camera to create a map of electrical activity in your sample, and you can do this in real time to see how electrical waves move across the heart. Here’s a simulation of electrical activity that Pras did with our Chaste software, and it includes a comparison with optical mapping at 39 seconds in the video below (this is for a completely different situation – just to show you how optical mapping works!):

Experimental Results

Greg found that in some intact hearts from mice recovering from ablation he could observe the optical mapping signal appearing to go straight through the middle of the scars. This is unexpected, to say the least, and required a lot of further investigation. I’ll let you read the full story in the paper, but suffice to say we confirmed that:

there aren’t any myocytes (normal heart muscle cells) left in the scars (shown by doing histology and using an electron microscope) and there are lots of what are largely fibroblasts left there, the cells in the scars are definitely ‘non myocytes’ anyway;

the observations aren’t just optical mapping artefacts (shown by doing direct micro-electrode recordings, showing that local electrical pulses from a suction electrode diffuse in, and even putting black foil in the middle of the hearts!);

the observations depend on electrical coupling between myocytes and fibroblasts, as it goes away if you don’t have this (shown by specifically breeding a strain of mice where you could knock out Connexin43 in just the non myocytes – clever stuff!). In fibroblasts, Connexin 43 is thought to form gap junctions electrically coupling them to only myocytes, not to other fibroblasts. If you knock out Connexin43 proteins in the non myocytes with this special mouse strain, then the optical mapping signal you record is much smaller in the scar region; the electrical signal no longer diffuses in from a suction electrode, indicating that fibroblast-myocyte junctions are required to see conduction into the scar.*

To quote from the article, this is “the first direct evidence in the intact heart that the cells in myocardial scar tissue are electrically coupled to the surrounding myocardium and can support the passive conduction of action potentials.”

Where mathematical modelling comes in

When Greg presented the experimental results at conferences he had a hard time convincing people that the experimental results were real, and not some odd artefacts of the experimental set up! It seemed that nobody expected electrical waves to travel through these regions, even if they were populated with cells, as these cells weren’t cardiac myocytes. It was counter-intuitive to most cardiac electrophysiologists that any signal would get across this gap, because that’s the whole point of ablation! This is where we thought it would be quite interesting to see what you would expect, from the standard model of electrical conduction, if there are neutral non-myocyte cells in a lesion that are coupled together. By neutral we mean not providing any ‘active’ transmembrane currents, and just providing passive electrical resistance.

Our standard simplest model for the reaction-diffusion of voltage in cardiac tissue is the monodomain equation (so-called because there’s an two-domain extension called the bidomain equation):

There are a few terms here that need defining: V is voltage across the cell membrane – which is the quantity that we consider to be diffusing around, rather than the individual types of ions themselves. Iion is the current across a unit surface area of the cell membrane at any point in space (varies across space), Istim is any external stimulus current which is applied to a unit volume of the tissue, Cm is the capacitance of a unit surface area of the membrane (the bigger this is, the more current is required to charge up a bit of membrane the same amount), and σ is the conductivity of the tissue – how easy it is for voltage to diffuse (expressed as a vector as there can be preferential conduction in different directions). χ is a scaling factor of surface-area-of-membrane-per-unit-volume-of-tissue, required to change the current across a unit-of-surface-area membrane into current across all-the-surface-area-of-membrane-in-a-unit-volume of tissue. So you can think of χ as the amount of membrane packed into a little cube of the tissue.

So, what will be different in the lesion?:

𝜎 , the cell-cell conductivity, could be varying. This would represent how much Connexin was present to link the lesion cells together with gap junctions.

𝜒 , the surface area of membrane in a unit volume of tissue, could also be varying. This would represent non-myocyte membrane density in the lesion.

𝐶𝑚, the capacitance of a unit of membrane area, probably won’t change (membrane made of same stuff). Beware though, a maths/biology language barrier means experimentalists might call 𝜒 ‘capacitance’ to confuse us all!

Iion will be small compared to myocytes, these cells aren’t actively trying to propagate electrical signals. So we did the study twice, once with this set to zero in the scar, and once with it set to use a previously published model of fibroblast electrophysiology (didn’t make any visual difference to the results).

Now in the scar region we aren’t applying an external stimulus so Istim = 0, and we can divide through by χ to get:

A nice property springs out – that the entire behaviour of the scar region (in terms of difference to the rest of the heart) is determined by the ratio of 𝜎/χ. So we introduce a factor ρ that will scale this quantity.

Even before we run any simulations, we’ve learnt something here by writing down the model and doing this “parameter lumping” (see nondimensionalisation for a framework in which to do this kind of thing rigorously!). Just by looking at ρ = 1 we see that there are an infinite number of ways we could get exactly the same behaviour. The scar cells could be just as well coupled (𝜎) and just as densely packed (χ) as the rest of the heart (incredibly unlikely to Greg), or they could be 1% as well coupled with 1% as much membrane present (more plausible to Greg); before we even do a simulation we can state definitively that this would give exactly the same behaviour: as ρ = 1 in both cases, and we’re solving the same equations! So this scaling is interesting, as we didn’t know whether the value of ρ would be increased or decreased in the scar, despite strong suspicion that the cells in the scar will have both reduced gap junctions and reduced membrane surface area available.

So we did a simulation with a mouse ventricular myocyte model, on a realistic sized bit of tissue 5mm x 5mm, with a 2mm diameter lesion in the middle. There are no preferential fibre directions here, something that could easily make the conduction more or less likely.

So what behaviours did we predict with different values of ρ? First for ρ = 1; the 𝜎/χ takes the same value as usual case (even though both could be reduced/increased by any factor):

In the video we see that conduction does indeed ‘get across’ the scar, even though it is not ‘driven’ as such as it is in the rest of the tissue, but instead is simply diffusing across that region. We predict the wave will get across a 2mm scar easily with this value of ρ.

What about with more membrane (or less conductivity): ρ = 0.1 (Remembering this would happen whenever 𝜎/χ is ten times less that the normal muscle: so both when 𝜎 is 1% of normal and χ is 10% of normal; or when 𝜎 is 0.1% of normal and χ is 1% of normal, for instance).

This time, you can see that the wave struggles to get across the lesion, but the membrane voltage still gets high, and that would probably still be high enough to record in optical mapping.

With less membrane density relative to the well-coupled-ness of the lesion cells (ρ = 10)? (Remembering this would happen whenever 𝜎/χ is ten times more that the normal muscle: so both when 𝜎 is 10% of normal and χ is 1% of normal; or when 𝜎 is 1% of normal and χ is 0.1% of normal, for instance). We see that the wave even appears to accelerate across the lesion region and conduction sets off earlier than before at the far side, as there is less membrane to charge so it is easier to do it:

So our conclusion was that it’s perfectly possible that a voltage signal could be recorded in the lesion, and a voltage signal could effectively travel straight through the scar, and conduction carry on out the other side. We’re not entirely sure about the value that ρ should take – but this behaviour was fairly robust, and matched what we saw in the experiments.

This simple model predicted many of the features of the recordings that were made from the scar region – see Figure 7 in the paper, and compare with experiments in Figure 4. So, it helped Greg answer accusations of “This is impossible!” that he got when he presented stuff at conferences, as he could reply that “If the cells have gap junctions, even without any voltage-gated ion channels of their own, this is exactly what we’d expect – see!”.

References

*Incidentally, I’d like to do a sociology experiment where I give biologists and mathematicians logic problems to do mentally. My hypothesis is that biologists would beat mathematicians, as they are always carrying around at least ten “if this then that” assumptions in their heads. Then I’d give them a pen and paper and see if the situation reversed…